IJMMS 2003:64, 4071–4083
PII. S0161171203206025
http://ijmms.hindawi.com
© Hindawi Publishing Corp.
A NOTE ON THE REDUCIBILITY OF LINEAR DIFFERENTIAL
EQUATIONS WITH QUASIPERIODIC COEFFICIENTS
XIAOPING YUAN and ANA NUNES
Received 3 June 2002
The system ẋ = (A+Q(t))x, where A is a constant matrix whose eigenvalues are
not necessarily simple and Q is a quasiperiodic analytic matrix, is considered. It
is proved that, for most values of the frequencies, the system is reducible.
2000 Mathematics Subject Classification: 37C55, 34A30, 34C20.
1. Introduction and results. Consider the quasiperiodic linear differential
equation
ẋ = A + Q(t) x
(1.1)
with x an n-dimensional vector, A a constant square matrix of order n, and
Q a square matrix of order n, quasiperiodic in time t. We say that a change
of variables x = P (t)y is a Lyapunov-Perron (LP) transformation if P (t) is
nonsingular and P (t), P −1 (t), and Ṗ (t) are bounded for all t ∈ R. Moreover, if P ,
P −1 , and Ṗ are quasiperiodic in time t, we refer to x = P (t)y as a quasiperiodic
LP transformation. If there is a quasiperiodic LP transformation x = P (t)y
such that y satisfies the equation
ẏ = By
(1.2)
with B a constant matrix, then we say that (1.1) is reducible.
The concept of the reducibility was first considered by Lyapunov (see [5]).
There are several authors who investigated the reducibility of (1.1) (see, e.g., [1,
2, 6]). The present paper complements the results obtained by Jorba and Simó
[2], which we will briefly recall. To this end, we will introduce some notation
and definitions that will be used throughout the paper.
We say that a function F is a quasiperiodic function in time t, with the
basic frequencies ω = (ω1 , . . . , ωr ), if there exists a function Ᏺ(θ1 , . . . , θr )
which is 2π -periodic in all its arguments θj , j = 1, . . . , r , and such that F (t) =
Ᏺ(ω1 t, . . . , ωr t). We call Ᏺ the hull of F (t). The function F will be called analytic quasiperiodic in a strip of width δ if, furthermore, Ᏺ is analytic in the
complex strip |Imθ| < δ.
4072
X. YUAN AND A. NUNES
Let λ = (λ1 , . . . , λn ) be the eigenvalues of A and λ0 () = (λ01 (), . . . , λ0n ()) the
eigenvalues of Ā := A + Q̄, where Q̄ is the average of Q(t),
1
T →∞ 2T
Q̄ = lim
T
−T
Q(t)dt.
(1.3)
Assume that Q(t) is analytic on the strip of width δ0 > 0 and that the vector
√
(λ, −1ω) satisfies the nonresonance conditions
−1k · ω + l · λ ≥
c
,
|k|γ
(1.4)
where l ∈ Zn with |l| = 0, 2 and 0 ≠ k ∈ Zr . It was shown by Jorba and Simó [2]
that (1.1) is reducible for in some Cantorian set Ᏹ ⊂ (0, 0 ), with 0 sufficiently
small, provided that
(1) the eigenvalues λ1 , . . . , λn of A are different;
(2) the eigenvalues λ01 (), . . . , λ0n () of Ā satisfy
d 0
0
d λi () − λj () > 2ρ > 0
(1.5)
=0
for some constant ρ and any 1 ≤ i < j ≤ n.
In [2], the basic idea is to kill the small perturbation Q(t) by KAM iteration.
Condition (2) is used to overcome the problem arising from the frequency
shift which comes up in this procedure. By a well-known theorem [3, pages
113–115], condition (1) guarantees that the eigenvalues λ0j () of Ā = A + Q̄
are differentiable in , and that therefore condition (2) can be imposed.
A natural question is: what happens when condition (1) or (2) is not satisfied?
The main result of the present paper is the following theorem which gives an
answer to this question.
Theorem 1.1. Let Ω0 ⊂ Rr+ be a compact set with positive Lebesgue measure
and assume that Q(t) is quasiperiodic with frequency ω ∈ Ω0 and analytic in
the strip of width δ0 > 0. Then, for a sufficiently small positive constant γ, there
2
exist a subset Ω ⊂ Ω0 with Meas(Ω0 \ Ω) = Meas(Ω0 )(1 − O(γ 1/n )) and a sufficiently small constant = (δ0 , γ) > 0 such that for any ∈ (0, ), system
(1.1) is reducible. More exactly, there is an analytic quasiperiodic transformation
x = P (t)y such that (1.1) is changed into
ẏ = By,
(1.6)
where B is a constant matrix with A − B = O().
The proof is based on the construction of an iterative lemma, Lemma 2.1.
In this construction, a finite number of terms in the Fourier expansion of the
perturbation are killed in each iteration, and the remainder is included in the
higher-order perturbation. The averaged perturbation is included in the timeindependent term. To solve the homological equation, avoiding the problem
A NOTE ON THE REDUCIBILITY OF LINEAR DIFFERENTIAL . . .
4073
of small divisors, certain frequencies must be removed from the original frequency set Ω0 at each iteration step. Showing that the remaining frequencies
form a big subset of Ω0 through the estimates of Section 3 concludes the proof.
Remark 1.2. When one of λj is not simple, the functions λ0j () are not necessarily differentiable in . Therefore, in the hypothesis of Theorem 1.1, we
have to regard the tangent frequencies ω, instead of , as the parameters used
to overcome the frequency shift in KAM iterative steps. Thus, we cannot find
explicitly a tangent frequency vector ω satisfying some Diophantine conditions such that Theorem 1.1 holds true. On the other hand, in Theorem 1.1 it
is not necessary to excise a subset of small measure from (0, ∗ ). In this sense,
Theorem 1.1 complements the results of [2]. Yet another complementary approach is that of [1], where ω is fixed and reducibility is proved for “most”
matrices A.
2. Proof of Theorem 1.1. The proof of Theorem 1.1 is based on Newton
iteration. Before we state the main iterative lemma, we need to introduce some
notation.
In the following, we denote by C, C1 , C2 , . . . positive constants which arise
in the estimates, by ᏽ the hull of a quasiperiodic function Q(t), and by ᏽ̃ the
average of ᏽ on the r -torus. For a matrix-valued function Q(t), define
QD := sup Q(t),
(2.1)
t∈D
where · is the sup-norm of the matrix.
Denote by m the number of the iterative step, and let
(1) m be the sequence that bounds the size of the perturbation before the
m−1
mth iteration step with m = (1+ρ)
and ρ = 1/3, for example;
(2) δm be the sequence that measures the size of the analyticity domain in
the angular variables after m iteration steps with
δm = δ0 −
∞
δ0 j −2
1 + · · · + m−2
2
j=1
for m ≥ 1;
(2.2)
(3) Um = U(δm ) = {θ ∈ (C/2π Z)N : | Im θ| < δm };
(4) qm be the sequence that measures the width of the analyticity domain
1/4n2
in the frequency space after m iteration steps with qm = m+1 , where
n is the dimensional number of system (1.1);
(5) C(m) be a constant of the form C1 mC2 .
Let Ω0 = Π0 ⊃ Π1 ⊃ · · · ⊃ Πm−1 be the closed sets in Rr+ and let Πm ⊂ Πm−1
be as defined inductively in Section 3. Let ᏻl be the complex ql -neighborhood
of Πl for l = 0, 1, . . . , m. Assume that, after m −1 steps of Newton iteration, we
4074
X. YUAN AND A. NUNES
get a quasiperiodic linear differential equation
ẋ = Am−1 + m Qm (t; ω) x,
(2.3)
where the following conditions are satisfied:
(H1)m Am−1 = A+1 ᏽ̃1 (ω)+· · ·+m−1 ᏽ̃m−1 (ω), m ≥ 2, A0 = A with ᏽ̃l (ω)
analytic in ᏻl , and ᏽ̃l ᏻl ≤ 1 for l = 1, . . . , m − 1;
(H2)m the hull ᏽm of Qm (t; ω) is analytic in Um−1 × ᏻm−1 and
ᏽm (θ, ω)Um−1 ×ᏻm−1 ≤ 1.
(2.4)
Am = Am−1 + m ᏽ̃m .
(2.5)
Let
Then (2.3) can be rewritten as
∗
(t) x,
ẋ = Am + m Qm
(2.6)
∗
(t) = Qm (t) − ᏽ̃m . Following [2], we will find a change of variables
where Qm
x = E + m Pm (t) y,
(2.7)
where E is the unit matrix such that (2.3) is changed into
ẋ = Am + O m+1 x
(2.8)
verifying conditions (H1)m+1 and (H2)m+1 . This change of variables is given by
the following lemma.
Lemma 2.1 (iterative lemma). Assume that (H1)m and (H2)m are fulfilled.
Then there is a quasiperiodic LP transformation
x = E + Pm (t) y,
(2.9)
where Pm (t) is quasiperiodic with frequency ω and its hull ᏼm (θ; ω) is analytic
in Um × ᏻm such that (2.3) is changed into
ẏ = Am + m+1 Qm+1 (t) y,
(2.10)
where Am and Qm+1 satisfy the conditions (H1)m+1 and (H2)m+1 .
Proof. Rewrite (2.3) as
∗
(t) x,
ẋ = Am + m Qm
(2.11)
∗
(t) = Qm (t) − ᏽ̃m and Am = Am−1 + ᏽ̃m . Hence, we can write
where Qm
∗
ᏽm
(θ, ω) =
0≠k∈Zr
∗
ᏽ̂m
(k; ω)e
√
−1k·θ
,
(2.12)
A NOTE ON THE REDUCIBILITY OF LINEAR DIFFERENTIAL . . .
4075
∗
∗
(k; ω) is the k Fourier coefficient of ᏽm
(θ, ω) in θ. Let Mm =
where ᏽ̂m
(1/bm )| ln m |, where bm = δm−1 − δm , and let
∗1
ᏽm
(θ, ω) =
∗
ᏽ̂m
(k; ω)e
0≠|k|≤Mm
∗2
ᏽm
(θ, ω) =
∗
ᏽ̂m
(k; ω)e
√
√
−1k·θ
−1k·θ
,
(2.13)
,
|k|>Mm
so that
∗
∗1
∗2
ᏽm
(θ, ω) = ᏽm
(θ, ω) + ᏽm
(θ, ω).
(2.14)
∗2
ᏽ (θ, ω)Um ×ᏻm ≤ C(m)m ,
m
(2.15)
We claim that
where C(m) is a constant of the form C1 mC2 . In fact,
√
ᏽ̂∗ (k; ω)ᏻm e −1k·θ Um
∗2
ᏽ (θ, ω)Um ×ᏻm ≤
m
m
|k|>Mm
ᏽ̂∗ (θ; ω)Um−1 ×ᏻm e−|k|δm−1 e|k|δm
≤
m
|k|>Mm
≤
e
−|k|(δm−1 −δm )
|k|>Mm
= m
(2.16)
e
−|k|(δm−1 −δm )
|k|>0
≤ C(m)m .
Next, we perform the change of variables as in (2.7), where E is the unit
matrix in Rn , to transform (2.10) into
ẏ =
−1 ∗1
E + m Pm
Am + m Am Pm − Ṗm + Qm
+ m+1 Qm+1 y,
(2.17)
where
−1 ∗
∗2
2
+ m
Q m Pm .
E + m Pm
m+1 Qm+1 = m Qm
(2.18)
We would like to have
−1 ∗1
E + m Pm
Am + m Am Pm − Ṗm + Qm
= Am
(2.19)
and this implies that
∗1
.
Ṗm = Am Pm − Pm Am + Qm
(2.20)
In order to solve this equation, we consider
ω·
∂ ᏼm
∗1
= Am ᏼm − ᏼm Am + ᏽm
,
∂θ
(2.21)
4076
X. YUAN AND A. NUNES
where ᏼ is the hull of P (t). Write
ᏼm (θ, ω) =
ᏼ̂m (k; ω)e
√
−1k·θ
.
(2.22)
0≠|k|≤Mm
Then we get
∗1
−1(k · ω)ᏼ̂m (k) = Am ᏼ̂m (k) − ᏼ̂m (k)Am + ᏽ̂m
(k),
0 < |k| ≤ Mm ,
(2.23)
where we omit the dependence on ω to simplify the notation. That is,
∗1
−1(k · ω)E − Am ᏼ̂m (k) + ᏼ̂m (k)Am = ᏽ̂m
(k),
0 < |k| ≤ Mm .
(2.24)
By Lemmas A.2 and 3.1, (2.24) is solvable for ω ∈ ᏻm and
ᏻ
−1 ᏻm ∗
ᏼ̂m (k; ω)ᏻm ≤ ᏽ̂m
−1k · ω En2 − En ⊗ Am + ATm ⊗ En
(k) m
C|k|τ ᏽ∗ Um−1 ×ᏻm−1 e−|k|δm−1
m
γm
C|k|τ ᏽm Um−1 ×ᏻm−1 e−|k|δm−1
≤
γm
C|k|τ −|k|δm−1
≤
e
,
γm
≤
(2.25)
where in the last inequality we have used (H2)m . Therefore,
ᏼm (θ, ω)Um ×ᏻm ≤
√
ᏼ̂m (k; ω)ᏻm e −1k·θ Um
0<|k|≤Mm
≤
≤
C|k|τ
e−|k|(δm−1 −δm )
γ
m
r
k∈Z
(2.26)
C 2
|k|τ e−C3 |k|(δ0 /m ) ≤ C(m),
γm k∈Zr
where the last inequality follows from Lemma A.1. Then, the function
Pm (t) = ᏼm ω1 t, . . . , ωr t; ω
(2.27)
solves (2.20). By (2.26) and (2.18), it is easy to show that ᏽm+1 Um ×ᏻm ≤ 1. We
omit the details.
Proof of Theorem 1.1. Obviously, (1.1) satisfies the conditions (H)m with
m = 1. In fact, condition (H2)1 may always be fulfilled by a suitable rescaling
of . Thus, by Lemma 2.1, there exists a sequence of transformations x = (E +
m Pm (t))y, m = 1, 2, . . . , such that the hulls ᏼm of the Pm (t) are analytic in
the domains Um × ᏻm , and
ᏼm Um ×ᏻm ≤ C(m).
(2.28)
A NOTE ON THE REDUCIBILITY OF LINEAR DIFFERENTIAL . . .
4077
Let
U∞ × ᏻ∞ =
∞
Um × ᏻm .
(2.29)
m=1
Then, all the ᏼm , m = 1, 2, . . . , are well defined in the domain U∞ × ᏻ∞ . Set
Φ(θ, ω) = · · · ◦ E + m ᏼm (θ, ω) ◦ · · · ◦ E + 2 ᏼ2 (θ, ω) ◦ E + 1 ᏼ1 (θ, ω) ,
Φ(t; ω) = · · · ◦ E + m Pm (t; ω) ◦ · · · ◦ E + 2 P2 (t; ω) ◦ E + 1 P1 (t; ω) .
(2.30)
Note that E +m ᏼm (θ; ω)U∞ ×ᏻ∞ ≤ 1+m C(m) ≤ 1+2−m . We see that Φ, and
thus Φ, are well defined. Let
x = Φ(t)y.
(2.31)
Since m ᏽm U∞ ×ᏻ∞ ≤ m → 0 as m → ∞, the transformation x = Φ(t)y
changes (1.1) into
ẏ = By,
(2.32)
∞
where B = A+ j=1 m ᏽ̃m . This, together with Lemma 3.3, completes the proof
of Theorem 1.1.
3. Estimates on the allowed frequencies set. Let Πl (0 ≤ l ≤ m − 1) be a
sequence of compact subsets of Rr+ with Ω = Π0 ⊃ Π1 ⊃ · · · ⊃ Πm−1 and denote
by ᏻl the complex ql -neighborhood of Πl , l = 0, . . . , m − 1. Recall that
Am (ω) = A + 1 ᏽ̃1 (ω) + · · · + m ᏽ̃m (ω),
(3.1)
where, for l = 1, . . . , m − 1, the ᏽ̃l (ω) are analytic, and real for real arguments
in the domain ᏻl , and ᏽ̃l (ω)ᏻl ≤ 1; and ᏽ̃m (ω) is analytic, and real for real
arguments in the domain ᏻm−1 , and ᏽ̃m (ω)ᏻm−1 ≤ 1. Denote by | · |d the
determinant of a matrix. Let
γm
᏾k (m) := ω ∈ Πm−1 : −1k · ω En2 − En ⊗ Am + ATm ⊗ En d <
,
|k|τ1
(3.2)
2
where γm = γ/m2n and τ1 = (r + 1)n2 ,
Πm = Πm−1 \
᏾k (m),
(3.3)
0<|k|≤Mm
where Mm = | ln m |/(δm−1 −δm ) is the number of Fourier coefficients we must
consider at the mth step of the iteration, and denote by ᏻm the complex qm neighborhood of Πm .
4078
X. YUAN AND A. NUNES
Lemma 3.1. Let τ = τ1 + n2 − 1 and
G(ω) =
−1k · ω En2 − En ⊗ Am (ω) + ATm (ω) ⊗ En .
(3.4)
Then, for ω ∈ ᏻm and 0 < |k| ≤ Mm , the inverse of G(ω) exists and it is analytic
in the domain ᏻm with
−1
G (ω)ᏻm ≤ Cγ −1 |k|τ .
m
(3.5)
Proof. By the definition of Πm , we get that for ω ∈ Πm and 0 < |k| ≤ Mm ,
ᏹk (ω) ≥ γm .
|k|τ1
(3.6)
It is easy to see that
G(ω)ᏻm ≤ C5 |k|,
k ≠ 0,
(3.7)
where C5 = 2(max{|ω| : ω ∈ Π} + A + 1). Since det G(ω) = ᏹk (ω), G−1 (ω)
exists for ω ∈ Πm and
G−1 (ω) =
adjG(ω)
,
ᏹk (ω)
(3.8)
where adj is the adjoint of a matrix. Thus, for 0 < |k| ≤ Mm ,
n2 −1
−1
−1
G (ω)Πm ≤ C6 |k|
= C6 γm
|k|τ .
γm /|k|τ1
(3.9)
Now, we assume that ω ∈ ᏻm . Then there is an ω0 ∈ Πm such that |ω − ω0 | <
qm . Thus,
−1 G
ω0 G(ω) − G ω0 Π ᏻ ≤ G−1 (ω) m ∇ω G(ω) m · ω − ω0 ᏻ
qm
−1
|k|τ G(ω) m−1
≤ C6 γm
qm−1 − qm
qm
−1
≤ C6 γm
|k|τ+1
qm−1 − qm
qm
τ+1 −1
≤ C6 Mm
γm
qm−1 − qm
τ+1 1/4n2 −1
2
C6 m(6+2r )n ln m m+1 γδ0
≤
1/4n2
1/4n2
m
− m+1
1
< .
2
(3.10)
A NOTE ON THE REDUCIBILITY OF LINEAR DIFFERENTIAL . . .
4079
Therefore, E + G−1 (ω0 )(G(ω) − G(ω0 )) has its inverse which is analytic in
ᏻm since
−1
E + G−1 ω0 G(ω) − G ω0
=
∞
j
− G−1 ω0 G(ω) − G ω0
.
(3.11)
j=0
So, G(ω) has its inverse for ω ∈ ᏻm and
−1
−1
G (ω) = E + G−1 ω0 G(ω) − G ω0
· G−1 ω0 −1 · G−1 ω0 ≤ E + G−1 ω0 G(ω) − G ω0
≤
(3.12)
−1
|k|τ .
Cγm
Lemma 3.2. Let K = n2 (n + 1)2 (diam Π0 )r −1 . Then the Lebesgue measure of
᏾k (m) verifies
2
Meas ᏾k (m) ≤
Kγ 1/n 1
.
|k|r +1 m2
(3.13)
1/4n2
Proof. Recall that ql = l+1 , let ql1 = (5/6)ql + (1/6)ql+1 , and denote
by ᏻ1l the complex ql1 -neighborhood of Πl . Obviously, ᏻl+1 ⊂ ᏻ1l ⊂ ᏻl and
dist(∂ ᏻ1l , ∂ ᏻl ) = (1/6)(ql − ql+1 ) > (1/12)ql . Noting that ᏽ̃l ᏻl ≤ 1 and using
Cauchy’s theorem, we get for 1 ≤ s ≤ n2 and 0 ≤ l ≤ m − 1,
s ᏻ1
s ᏻ
1/2
l ∂ω
ᏽ̃l l ≤ l 12ql−1 ᏽ̃l l ≤ l .
(3.14)
The combination of (3.1) and (3.14) leads to
1
s
∂ Am (ω)ᏻm−1 ≤ 1/2 .
ω
(3.15)
B(ω) := −En ⊗ Am (ω) + ATm ⊗ En .
(3.16)
1
s
∂ B(ω)ᏻm−1 ≤ 1/2 .
ω
(3.17)
ᏹk (ω) = −1k · ω En2 + B(ω)d .
(3.18)
Let
Then
Set
s
ᏹk . To this end, write B(ω) = (bij ).
We are now in a position to estimate ∂ω
Then
n2
2
2
ᏹk (ω) = −1 (k · ω)n +
φl (ω)(k · ω)n −l ,
(3.19)
1≤l≤n2 −1
4080
X. YUAN AND A. NUNES
where
φl (ω) =
σj1 ···jl b1j1 · · · bljl
(3.20)
1≤ji ≤n2
√
√
and σj1 ···jl ∈ {−1, +1, − −1, + −1}.
Observe that for 1 ≤ l ≤ n2 and ω ∈ ᏻ1m−1 ,
1
s
∂ bij ≤ ∂ s B(ω)ᏻm−1 ≤ 1/2 .
ω1
ω
(3.21)
Thus, for ω ∈ ᏻ1m−1 and 1 ≤ s ≤ n2 ,
s
d ·
·
·
b
b
ljl ≤ dωs 1j1
ds1
dsl
b
b
·
·
·
s1 1j1
sl 1jl dω
dω
1
s1 +···+sl =s
1
1
≤ (1/2)(s1 +···+sl )
1
(3.22)
s1 +···+sl =s
s
≤ 21/2 ,
and therefore,
2
s
d
n 1/2 s
2
.
dωs φl (ω) ≤ l
(3.23)
1
Without loss of generality, assume that |k| = |k1 | + · · · + |kr | ≤ r |k1 |. Hence,
for every ω ∈ ᏻ1m−1 ,
n2
d
2
dωn
1
n2 −l
φl (ω)(k · ω)
1≤l≤n2 −1
dn2 n2 −l φl (ω)(k · ω)
≤
dωn2
1
1≤l≤n2 −1
2
dn2 −s
n2 ds
n2 −l φ
(ω)
(k
·
ω)
≤
l
2
2
s
n
−s
dω
s dω1
1
1≤l≤n2 −1 l≤s≤n2
2
n2
n2 1/2 s k1 n −s |k · ω|s−l n2 !
2
≤
l
s
2
2
(3.24)
1≤l≤n −1 l≤s≤n
n2 −1 2
n !,
≤ C4 1/2 k1 where C4 is some constant which depends only on n, r , and on the maximum
of |ω| in Π0 . Obviously,
dn
2
2
dωn
1
n 2
2
(k · ω)n = n2 !k1 .
(3.25)
4081
A NOTE ON THE REDUCIBILITY OF LINEAR DIFFERENTIAL . . .
Thus, in ᏻ1m−1 , we have
dn2
n 2 −1 1 2 n2
≥ n2 !k1 1 − C4 1/2 k1 > n !k1 ᏹ
(ω)
dωn2 k
2
1
(3.26)
provided that is small enough so that C4 1/2 < 1/2. Using (3.26) and Lemma
A.3, we get
Meas ᏾k (m) ≤ n n2 + 1
2
γm
|k|τ1
1/n2
diam Π0
r −1
2
(3.27)
Kγ 1/n
1
≤
·
.
|k|r +1 m2
This completes the proof.
By Lemma 2.1, the nested sequence of closed sets
Ω0 = Π 0 ⊃ Π 1 ⊃ · · · ⊃ Π m ⊃ · · ·
(3.28)
is defined inductively. The following lemma is a corollary of Lemma 3.2.
Lemma 3.3. Let
Π∞ =
∞
Πm .
(3.29)
m=0
2
Then Meas Π∞ = (Meas Π0 )(1 − O(γ 1/n )).
Appendix
Lemma A.1. For δ > 0 and ν > 0, the following inequality holds true:
k∈ZN
e−2|k|δ |k|ν ≤
ν
1
ν
(1 + e)N .
e
δν+N
(A.1)
Proof. This lemma can be found in [1]. We will find the value of z ≥ 1
yielding a maximum value for the expression ν ln z − δz. Differentiating it in
z and equating the result to zero, we get that ν/z = δ and z = ν/δ > 1. From
this it follows that
ν
ν ln z − δz ≤ ν ln − 1 .
δ
(A.2)
ν
ν
ν
exp(δz)
=
.
zν ≤ exp(δz) exp ν ln − 1
δ
e
δν
(A.3)
This expression yields
4082
X. YUAN AND A. NUNES
Thus,
k∈ZN
ν
ν
1 −|k|δ
e
e
δν k
ν
ν
1 1 + exp(−δ) N
=
e
δν 1 − exp(−δ)
ν
ν
1 1+e N
≤
.
ν
e
δ
δ
e−2|k| |k|ν ≤
(A.4)
Lemma A.2. Let A, B, and C be r × r , s × s, and r × s matrices, respectively;
and let X be an r × s unknown matrix. Then the matrix equation
AX + XB = C
(A.5)
is solvable if and only if the vector equation
E s ⊗ A + B T ⊗ Er X = C (A.6)
is solvable, where X = (X1T , . . . , XsT )T , C = (C1T , . . . , CsT )T if we write X = (X1 , . . . ,
Xs ) and C = (C1 , . . . , Cs ). Moreover,
−1 C
X ≤ Es ⊗ A + B T ⊗ Er
(A.7)
if the inverse exists.
Proof. This lemma can be found in many textbooks on matrix theory; for
example, [4, page 256].
The following lemma can be found in [7, page 23].
Lemma A.3. Let Ᏽ be an interval in R1 and Ᏽ̄ its closure. Suppose that g : Ᏽ̄ →
C is k times continuously differentiable. Let Ᏽh = {x ∈ Ᏽ̄ : |g(x)| ≤ h}, h > 0. If,
for some constant d > 0, |dk g(x)/dx k | ≥ d for any x ∈ Ᏽ, then Meas Ih ≤ ch1/k
with c = 2(2 + 3 + · · · + k + d−1 ).
Acknowledgment. This work was carried out during the first author’s
stay at the Centro de Matemática e Aplicações Fundamentais (CMAF), Universidade de Lisboa with a Postdoctoral Fellowship Grant from the Fundação para
a Ciência e Tecnologia.
References
[1]
[2]
N. N. Bogoljubov, Ju. A. Mitropoliskii, and A. M. Samoı̆lenko, Methods of Accelerated Convergence in Nonlinear Mechanics, Springer-Verlag, New York, 1976,
translated from the original Russian, Naukova Dumka, Kiev, 1969.
À. Jorba and C. Simó, On the reducibility of linear differential equations with
quasiperiodic coefficients, J. Differential Equations 98 (1992), no. 1, 111–
124.
A NOTE ON THE REDUCIBILITY OF LINEAR DIFFERENTIAL . . .
[3]
[4]
[5]
[6]
[7]
4083
T. Kato, Perturbation Theory for Linear Operators, corrected printing of the 2nd
ed., Springer-Verlag, Berlin, 1980.
P. Lancaster, Theory of Matrices, Academic Press, New York, 1969.
V. V. Nemytskii and V. V. Stepanov, Qualitative Theory of Differential Equations,
Princeton Mathematical Series, no. 22, Princeton University Press, New Jersey, 1960.
J. Xu and Q. Zheng, On the reducibility of linear differential equations with
quasiperiodic coefficients which are degenerate, Proc. Amer. Math. Soc. 126
(1998), no. 5, 1445–1451.
J. You, Perturbations of lower-dimensional tori for Hamiltonian systems, J. Differential Equations 152 (1999), no. 1, 1–29.
Xiaoping Yuan: Department of Mathematics, Fudan University, Shanghai 200433,
China
Current address: Centro de Matemática e Aplicações Fundamentais, Universidade de
Lisboa, 1649-003 Lisboa, Portugal
Ana Nunes: Centro de Matemática e Aplicações Fundamentais, Universidade de Lisboa, 1649-003 Lisboa, Portugal
E-mail address: anunes@lmc.fc.ul.pt